The Method of False Position

Date: 02/06/2004 at 23:37:09
From: Lynne
Subject: Math Riddle
There is a quantity such that 2/3 of it, 1/2 of it, and 1/7 of it
added together becomes 33. What is the quantity? Solve the problem
by the method of false position.
I know that using the method of false position, we are looking for
the root of an equation, and need a way of making a guess that is
better than our previous guess.

Date: 02/07/2004 at 14:59:35
From: Doctor Douglas
Subject: Re: Math Riddle
Hi Lynne -
Thanks for writing to the Math Forum.
Let's let 'x' be the unknown quantity, and we can write the following
equation:
x*(2/3) + x*(1/2) + x*(1/7) = 33
Assume x = 42. This is a convenient first guess because it is
divisible by all of the denominators {3,2,7}, which makes our life a
bit easier on the first step.
42*(2/3) + 42*(1/2) + 42*(1/7) = 28 + 21 + 6 = 55.
So our first guess of 42 is approximately too big by a factor of
55/33 = 5/3. Our next guess is therefore
42*(3/5) = 126/5 or 25.2.
We plug this, our second guess, in for x in the equation above
to obtain
(126/5)*(2/3) + (126/5)(1/2) + (126/5)(1/7) = 84/5 + 63/5 + 18/5
= 165/5 = 33
which is exactly what we want it to be, and we've therefore found
the value of x for which the equation is true. There is of course a
more direct method to solve the original equation using algebra
techniques, but I think that this problem shows you how the method of
false position works so that you can also apply it to cases where you
cannot simply solve for x.
- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/